Hilbert series for leptonic flavor invariants in the minimal seesaw model

Abstract

In this paper, we examine the leptonic flavor invariants in the minimal see-saw model (MSM), in which only two right-handed neutrino singlets are added into the Standard Model in order to accommodate tiny neutrino masses and explain cosmological matter-antimatter asymmetry via leptogenesis mechanism. For the first time, we calculate the Hilbert series (HS) for the leptonic flavor invariants in the MSM. With the HS we demonstrate that there are totally 38 basic flavor invariants, among which 18 invariants are CP-odd and the others are CP-even. Moreover, we explicitly construct these basic invariants, and any other flavor invariants in the MSM can be decomposed into the polynomials of them. Interestingly, we find that any flavor invariants in the effective theory at the low-energy scale can be expressed as rational functions of those in the full MSM at the high-energy scale. Practical applications to the phenomenological studies of the MSM, such as the sufficient and necessary conditions for CP conservation and CP asymmetries in leptogenesis, are also briefly discussed.

CP violation and flavor invariants in the seesaw effective field theory

“…This result has been partly derived in Ref [50]. for the minimal seesaw model, but the inclusion of the dimension-six operator as well as a complete matching is lacking therein.…”

“…On the other hand, the algebraic structure of the invariant ring and the construction of flavor invariants in the full seesaw model have been partly studied in Refs. [28,29,50]. Thus an intriguing question is what the connection between the invariant ring of the flavor space in the SEFT and that in the full theory is and how the two sets of flavor invariants match with each other.…”

“…The plethystic program [42] provides a convenient method to find out all the basic invariants as well as the polynomial identities (known as syzygies) among them. In the type-I seesaw model [43][44][45][46][47] and its low-energy effective theory with only the dimension-five Weinberg operator [48], the HS in the flavor space have been calculated and the basic flavor invariants are explicitly constructed [28,29,49,50]. Moreover, the renormalization-group equations of all the basic flavor invariants in the effective theory have been derived as well [49].…”

CP violation and flavor invariants in the seesaw effective field theory

“…The plethystic program [43] provides a convenient method to find out all the basic invariants as well as the polynomial identities (known as syzygies) among them. In the type-I seesaw model [44][45][46][47][48] and its low-energy effective theory with only the dimension-five Weinberg operator [49], the HS in the flavor space have been calculated and the basic flavor invariants are explicitly constructed [28,29,50,51]. Moreover, the renormalization-group equations of all the basic flavor invariants in the effective theory have been derived as well [50].…”

Beyond Jarlskog: 699 invariants for CP violation in SMEFT

“…[11,[21][22][23] for the case of additional generations of matter, refs. [24][25][26][27][28][29][30][31][32] for the inclusion of neutrino masses, ref. [33] for vector-like extensions, ref.…”